Precalculus Examples

$\cos^{2} (x) + \tan^{2} (x) \cos^{2} (x) = 1$

Step 1

Replace the $\cos^{2} (x)$ with $1 - \sin^{2} (x)$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$(1 - \sin^{2} (x)) + \tan^{2} (x) \cos^{2} (x) = 1$

Step 2

Simplify the left side.

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Step 2.1

Simplify $(1 - \sin^{2} (x)) + \tan^{2} (x) \cos^{2} (x)$ .

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Step 2.1.1

Apply pythagorean identity.

$\cos^{2} (x) + \tan^{2} (x) \cos^{2} (x) = 1$

Step 2.1.2

Factor $\cos^{2} (x)$ out of $\cos^{2} (x) + \tan^{2} (x) \cos^{2} (x)$ .

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Step 2.1.2.1

Multiply by $1$ .

$\cos^{2} (x) \cdot 1 + \tan^{2} (x) \cos^{2} (x) = 1$

Step 2.1.2.2

Factor $\cos^{2} (x)$ out of $\tan^{2} (x) \cos^{2} (x)$ .

$\cos^{2} (x) \cdot 1 + \cos^{2} (x) \tan^{2} (x) = 1$

Step 2.1.2.3

Factor $\cos^{2} (x)$ out of $\cos^{2} (x) \cdot 1 + \cos^{2} (x) \tan^{2} (x)$ .

$\cos^{2} (x) (1 + \tan^{2} (x)) = 1$

Step 2.1.3

Rearrange terms.

$\cos^{2} (x) (\tan^{2} (x) + 1) = 1$

Step 2.1.4

Apply pythagorean identity.

$\cos^{2} (x) \sec^{2} (x) = 1$

Step 2.1.5

Rewrite $\sec (x)$ in terms of sines and cosines.

$\cos^{2} (x) {(\frac{1}{\cos (x)})}^{2} = 1$

Step 2.1.6

Simplify the expression.

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Step 2.1.6.1

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\cos^{2} (x) \frac{1^{2}}{\cos^{2} (x)} = 1$

Step 2.1.6.2

One to any power is one.

$\cos^{2} (x) \frac{1}{\cos^{2} (x)} = 1$

Step 2.1.7

Cancel the common factor of $\cos^{2} (x)$ .

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Step 2.1.7.1

Cancel the common factor.

$\cos^{2} (x) \frac{1}{\cos^{2} (x)} = 1$

Step 2.1.7.2

Rewrite the expression.

$1 = 1$

Step 3

Since $1 = 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 4

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$